Arc Length

Understanding Arcs in Circles

• A semicircle is defined as half of a circle.

• An arc is part of the circumference of a circle that lies between two points.

Examples of Arcs

• For instance, if point K is on circle P, any portion of the circumference between points is an arc (represented as arc YKM).

• You can have multiple arcs in a circle, and each is defined by its endpoints on the circle.

Analyzing Arc Lengths

• To find what fraction of the circumference arc YKM represents, one can often simplify ratios involving angles compared to the entire circle (360 degrees).

• For example, a central angle leads to a relationship with the entire circumference:

  • If arc W-C is a sixth of the circle, it means if divided into six equal slices, W-C represents one slice.

Key Takeaways about Arc Length

• Arc length is the distance along the circle's circumference between two points.

• To relate arc length to angles, remember that the measure of an arc is proportional to the ratio of the angle to the whole circle (360 degrees).

Proportional Relationships

• If two circles (one smaller, one larger) share the same central angle, even if their radii differ, their arc measures will be the same.

• This means that the proportions associated with arc lengths relative to the whole circle are also consistent across circles with varying sizes—same central angle leads to same arc ratio.

Arc Length Formula

• Record the Arc Length Formula in your notes for later use:

  • L = (θ/360) × (2πr)

  • Where:

    • L = arc length

    • θ = central angle in degrees

    • r = radius of the circle

    • π is approximately 3.14

Practical Application: Ferris Wheel Problem

• In analyzing Dominic's ride on a Ferris wheel, for instance:

  • The radius is 60 feet.

  • The wheel is divided into 18 segments (seats), and after traveling from point A to point B, you need to calculate how far Dominic traveled.

  • This involves finding the total circumference first, then calculating arc length.

Steps for Calculation

1. Calculate Circumference: C = 2πr where r = 60 feet

2. Determine Fraction of Arc: Count segments from A to B; if 6 out of 18, then you have 1/3 of the arc

3. Find Arc Length: Multiply circumference by 1/3 to get the arc length

4. For Multiple Revolutions: If he rotates 4 times, then multiply derived arc length by 4 to find total distance traveled

• Example result: Arc length might equal 40π feet for one complete segment, and total distance would adjust accordingly based on the number of revolutions.